Josephson-based threshold detector for Lévy distributed fluctuations
We propose a threshold detector for Lévy distributed fluctuations based on a Josephson junction. The Lévy noise current added to a linearly ramped bias current results in clear changes in the distribution of switching currents out of the zero-voltage state of the junction. We observe that the analysis of the cumulative distribution function of the switching currents supplies information on both the characteristics shape parameter of the Lévy statistics and the intensity of the fluctuations. Moreover, we discuss a theoretical model which allows from a measured distribution of switching currents to extract characteristic features of the Lévy fluctuations. In view of this results, this system can effectively find an application as a detector for a Lévy signal embedded in a noisy background.
Introduction. – A current-biased Josephson junction (JJ) represents a natural threshold detector for current fluctuations, inasmuch as it is a metastable system operating on an activation mechanism. Actually, the behaviour of a JJ can be depicted as a particle, representing the superconducting phase difference across the JJ, in a cosine “washboard” potential with friction Barone and Paternò (1982); Likharev (1986), see Fig. 1(a). In this picture, the slope of the washboard potential is given by the injected current, and the dynamics of the phase is described by the resistively and capacitively shunted junction (RCSJ) model. The equivalent particle remains near a washboard minimum (correspondingly, the JJ is in the zero-voltage metastable state) until the direct bias current exceeds a critical value, or a fluctuation sets the phase in motion along the potential. Indeed, a current fluctuation instantaneously tilts the potential, and a noise-induced escape from a minimum can occur. In correspondence of the escape a voltage develops, as the voltage is related to the velocity of the phase particle. Shortly, if a JJ is set in the fundamental zero-voltage state, noise can cause a passage from this zero-voltage state to the finite voltage “running” state. The statistics of these passages can be exploited to reveal the features of the noise.
After early suggestions Pekola (2004); Ankerhold and Grabert (2005); Tobiska and Nazarov (2004), several proposals for concrete experimental setup of Josephson-based noise detectors have been put forward Pekola et al. (2005); Ankerhold (2007); Sukhorukov and Jordan (2007); Peltonen et al. (2007); Huard et al. (2007); Grabert (2008); Urban and Grabert (2009). A scheme to detect the Poissonian character of the charge injection in an underdamped JJ, based on the analysis of the third-order moment of the electrical noise, was proposed in Ref. Pekola (2004) and a scheme to detect the fourth-order moment of the noise was discussed in Ref. Ankerhold and Grabert (2005). A threshold detector based on an array of overdamped JJs for the direct measurement of the full counting statistics, through rare over-the-barrier jumps induced by current fluctuations, was suggested in Ref. Tobiska and Nazarov (2004). Alternatively, in the Coulomb blockade regime, the sensitivity of the JJ conductance to the non-Gaussian character of the applied noise was demonstrated Lindell et al. (2004); Heikkilä et al. (2004). Most proposals make use of the information content of higher moments, beyond the variance, of the electric noise, mainly to discuss the Poissonian character of the current fluctuations. However, experimental measurements of third and fourth moments are actually demanding and error-prone with respect to measurements of dc-transport properties. Indeed, deviations from Gaussian behaviour are typically small and high frequency regimes are usually necessary to retrieving information about fluctuations.
In this Letter we address the issue of Lévy distributed noise characterization through the switching currents distribution (SCD) of a JJ Addesso et al. (2012, 2013). These stochastic processes could drive the particle, namely, the phase in JJ context, over a very long distance in a single motion event, namely, a flight. Lévy flights well describe transport phenomena in different condensed matter systems Bardou et al. (1994); Woyczyński (2001); Pereira et al. (2004); Novikov et al. (2005); Barthelemy et al. (2008); Augello et al. (2010); Luryi et al. (2012); Semyonov et al. (2012); Briskot et al. (2014); Subashiev et al. (2014); Vermeersch et al. (2014); Mohammed et al. (2015); Vermeersch et al. (2015a, b); Gattenlöhner et al. (2016); Gutman et al. (2016). Results on Lévy flights were recently reviewed in Refs. Dubkov et al. (2008); Zaburdaev et al. (2015) and an extensive bibliography on -stable distributions is maintained online by Nolan Nolan (2017). To visualize the effect of Lévy noise, in Fig. 1(b) we show several phase trajectories, in the absence of bias current, characterized by abrupt fluctuations. A Lévy flights distribution exhibits power-law tails and, consequently, second and higher moments diverge. The latter feature poses a relevant complication in relating Lévy flight models to experimental data. The problem is to accurately perform the experimental measurement of a physical quantity that, according to a possibly infinite variance, can suffer limitless fluctuations. A JJ-based threshold detector circumvents this difficulty, since the switching occurs as the phase particle passes a potential barrier, regardless the intensity of the fluctuation. The distribution of the current values in correspondence of which a switch occurs, i.e., the switching currents , catches the information content we are interested in. Markedly, the investigation of SCDs paves the way for the direct experimental investigation of an -stable Lévy noise signal or a Lévy component of an unknown noise signal.
Model. – A typical setup for a Josephson-based noise readout, e.g., Refs. Pekola et al. (2005); Sukhorukov and Jordan (2007); Urban and Grabert (2009), consists of a JJ on which both a bias current drawn from a parallel source, , and a stochastically fluctuating current, , are flowing [see Fig. 1(c)]. We shall not discuss here escapes guided by macroscopic quantum tunnelling Grabert and Weiss (1984), occurring at very low temperatures, and we consider exclusively processes activated by noise fluctuations.
In this readout scheme, the noise influence is considered in the limit of adiabatic bias regime, where the change of the potential slope induced by the bias current is slow enough to keep the phase particle in the metastable well until the noise pushes out the particle. A measurement consists in slowly and linearly ramping the bias current in a time , so that ( is the critical current of the JJ), and to record the value at which a switch occurs. In this work, sequences of ramps of maximum duration are applied, where and are the plasma frequency and the capacitance of the JJ, respectively. Finally, a SCD is obtained.
The phase dynamics is obtained by numerically solving the RCSJ model equation Barone and Paternò (1982)
where is the flux quantum and is the normal resistance of the JJ. Moreover, is the washboard potential [see Fig. 1(a)]
where . The average slope of the potential is given by , where is the ramp speed. The resulting activation energy barrier confines the phase in a potential minimum.
Eq. (1) can be recast for convenience in a compact form
where is the effective junction mass, the friction is governed by the parameter , , and is the stochastic term. In these units, . In all simulations we assume the damping , the ramp speed , and the Lévy noise intensity .
To model the Lévy noise sources, we use the algorithm proposed by Weron Weron (1996) for the implementation of the Chambers method Chambers et al. (1976). The notation is used for the Lévy distributions Dubkov and Spagnolo (2007); Valenti et al. (2014); Guarcello et al. (2016), where is the stability index, is called asymmetry parameter, and and are a scale and a location parameter, respectively. The stability index produces the asymptotic long-tail power law for the distribution, which for is of the type, while gives the Gaussian distribution. We consider exclusively symmetric (i.e., with ), bell-shaped, standard (i.e., with and ), stable distributions , with .
Lévy escape. – Escapes over a barrier in the presence of Lévy noise have been thoroughly investigated for the overdamped case Dybiec et al. (2006, 2007); Dubkov et al. (2008, 2009). If both the distance between neighbour minimum and maximum of a metastable potential and the height of the potential barrier [see Fig. 1(a)] are unitary ( and , respectively), the power-law asymptotic behaviour of the mean escape time for the Lévy statistics reads Chechkin et al. (2005, 2007)
where both the power-law exponent and the coefficient depend on . For arbitrary spatial and energy scale, by rescaling time, energy, and space in the overdamped case of Eq. (3) Chechkin et al. (2005), Eq. (4) is replaced by
The scaling exponent and the coefficient are supposed to have a universal behaviour for overdamped systems, in particular Chechkin et al. (2005). Then, by assuming in the prefactor, Eq. (5) becomes Chechkin et al. (2005, 2007); Dubkov et al. (2008)
The physical interpretation of the previous assumption is that in the presence of Lévy flights the mean escape time is independent on and only depends upon . The escape rate is inversely proportional through the coefficient (see Eq. (6)) to the noise parameters .
Lévy noise induced switching currents. –
The switching currents , are the experimental evidence of escape processes in JJs. A collection of escapes can be characterized by a probability distribution function (PDF) of switching currents, as shown in Fig. 2(a) for three peculiar cases, . For the lowest value, i.e., , the PDF resembles an exponential distribution. For , i.e., the Cauchy-Lorentz distribution, the PDF is roughly flat. Finally, for , the distribution approaches the PDF of a Gaussian noise. The most evident distinction between Lévy and Gaussian cases lies in the low currents behaviour of PDFs: in the former case, the switching probability is sizable, while in the latter case it is vanishingly small Fulton and Dunkleberger (1974).
The effect of the parameter is further elucidated in the cumulative distribution functions (CDFs), namely, the probability that takes a value less than or equal to , shown in Fig. 2(b). Here, we note that each CDF at a given value of decreases with . Therefore, CDFs are suitable for the estimation of Guarcello et al. (2016). To model the CDFs of the switching currents, we exploit Eq. (6) to describe the escape rates over a barrier. The average escape times estimated by Eq. (6) allow to connect the switching currents with the properties of the Lévy noise. The CDF of as function of for a specific initial value of the bias ramp, , reads
for the PDF associated to Eq. (7) as a function of the average escape time (here is an appropriated normalization constant). For the thermal noise, Kramers’ formula entails that escapes across the barrier depend on the barrier height. For Lévy noise, with the same widely employed approximations behind Eq. (6), turns out independent of the barrier height , and becomes only function of , see Eq. (2). The expression of , Eq. (6), inserted in Eq. (8) gives for the Lévy statistics (at the first order in )
This is a further step forward with respect to results of Ref. Guarcello et al. (2017), concerning the nonsinusoidal potential appropriated for graphene JJs Guarcello et al. (2015); Giubileo et al. (2017); Spagnolo et al. (2017); Lee and Lee (2017), inasmuch as the above equation contains the explicit expression for the argument of the exponential.
Notably, the solution of Eq. (8) can be analytically computed and expressed in a compact form by using the function defined as
where is the exponential integral with argument Prudnikov et al. (1998). Then, the PDF can be written as
The corresponding CDF is
This is the main result of this work, that is to connect the properties of Lévy flights with the accessible quantity of SCDs. It is important to remind the main approximations underlying Eq. (13): it has been assumed that the result obtained for an overdamped system, see Eq. (6), still holds for moderately underdamped systems, and that Eq. (8), which is strictly valid in the adiabatic regime, can be applied to a slowly varying process.
We have performed extensive numerical simulations to check the validity of results given by Eqs. (6) and (13). In Fig. 3 we show the marginal CDF, i.e., restricted to the maximum bias , for and . The choice of these values for and arises from practical considerations, since Eqs. (6) and (13) are more accurate for low bias currents and low values, respectively. For these values the Lévy flight jump features dominate, while in the opposite limits, and , the Gaussian characteristics set in. Accordingly, in the considered range of values the effects of the Gaussian noise contribute can be safely ignored. The main panel of Fig. 3 shows also the numerical curves obtained by fitting of Eq. (13). The agreement between computational results and the theoretical analysis, see Eq. (13), is quite accurate for . For the statistics of switches becomes undistinguishable from the uniform distribution (the bisector in Fig. 3). Thus, the model we proposed can be used to determine from switching currents measurements (as the other parameters are known), but it proves to be especially valuable for .
In the inset of Fig. 3 we show with red circles the estimate of the coefficient obtained by numerical fitting of Eq. (13) of the marginal CDFs shown in the main panel. The estimates of the values of significantly deviate from both the numerical estimates given in Ref. Chechkin et al. (2007) and the analytical estimate obtained in Ref. Chechkin et al. (2005). However, these differences can be ascribed to: i) an overdamped rather than underdamped dynamics; ii) a fixed rather than a slowly varying potential barrier; iii) a cubic rather than a cosine potential.
Conclusions. – We have investigated the switching currents distributions (SCDs) in Josephson junctions in the presence of a Lévy noise source. Lévy distributed fluctuations are characterized by scale-free jumps or Lévy flights. Consequently, we expect the SCDs to exhibit a peculiar behaviour markedly different from the Gaussian noise case. The aim is to detect the characteristics of the Lévy noise from SCDs. Specifically, depending on the value of the stability distribution index, , we have numerically found that: i) for , the SCDs are peaked at zero bias current; ii) for , the SCD is roughly flat; iii) finally, for , the SCDs are peaked at high bias currents (alike the usual Gaussian noise induced peak) and slowly decrease at low bias currents. A peculiar behaviour can be observed also in the cumulative distribution function (CDF) curves, that at a given value of decrease with increasing . Moreover, CDFs are convex for , and concave for (the case corresponds to a linear CDF).
A theoretical good estimate of the SCDs can be retrieved on the basis of the Fulton adiabatic approach Fulton and Dunkleberger (1974) and assuming that the average escape time for the Lévy guided overdamped case can be extended to moderately damped systems. These theoretical findings are confirmed by the abovementioned numerical observations. Moreover, the theoretical approach recovers a previous result Guarcello et al. (2017), where a phenomenological linear approximation has been applied [see Eq.(9)]. Finally, we achieve, from the SCDs through the theoretical model [see Eqs. (11) and (13)] the estimate of the universal (i.e., barrier height independent) noise coefficient and then, if the other parameters are known, the value of the stability index .
- Barone and Paternò (1982) A. Barone and G. Paternò, Physics and Applications of the Josephson Effect (Wiley, New York, 1982).
- Likharev (1986) K. Likharev, Dynamics of Josephson Junctions and Circuits (Gordon & Breach, New York, 1986).
- Pekola (2004) J. P. Pekola, Phys. Rev. Lett. 93, 206601 (2004).
- Ankerhold and Grabert (2005) J. Ankerhold and H. Grabert, Phys. Rev. Lett. 95, 186601 (2005).
- Tobiska and Nazarov (2004) J. Tobiska and Y. V. Nazarov, Phys. Rev. Lett. 93, 106801 (2004).
- Pekola et al. (2005) J. P. Pekola, T. E. Nieminen, M. Meschke, J. M. Kivioja, A. O. Niskanen, and J. J. Vartiainen, Phys. Rev. Lett. 95, 197004 (2005).
- Ankerhold (2007) J. Ankerhold, Phys. Rev. Lett. 98, 036601 (2007).
- Sukhorukov and Jordan (2007) E. V. Sukhorukov and A. N. Jordan, Phys. Rev. Lett. 98, 136803 (2007).
- Peltonen et al. (2007) J. Peltonen, A. Timofeev, M. Meschke, T. Heikkilä, and J. Pekola, Physica E (Amsterdam) 40, 111 (2007).
- Huard et al. (2007) B. Huard, H. Pothier, N. Birge, D. Esteve, X. Waintal, and J. Ankerhold, Ann. Phys. 16, 736 (2007).
- Grabert (2008) H. Grabert, Phys. Rev. B 77, 205315 (2008).
- Urban and Grabert (2009) D. F. Urban and H. Grabert, Phys. Rev. B 79, 113102 (2009).
- Lindell et al. (2004) R. K. Lindell, J. Delahaye, M. A. Sillanpää, T. T. Heikkilä, E. B. Sonin, and P. J. Hakonen, Phys. Rev. Lett. 93, 197002 (2004).
- Heikkilä et al. (2004) T. T. Heikkilä, P. Virtanen, G. Johansson, and F. K. Wilhelm, Phys. Rev. Lett. 93, 247005 (2004).
- Addesso et al. (2012) P. Addesso, G. Filatrella, and V. Pierro, Phys. Rev. E 85, 016708 (2012).
- Addesso et al. (2013) P. Addesso, V. Pierro, and G. Filatrella, Europhys. Lett. 101, 20005 (2013).
- Bardou et al. (1994) F. Bardou, J. P. Bouchaud, O. Emile, A. Aspect, and C. Cohen-Tannoudji, Phys. Rev. Lett. 72, 203 (1994).
- Woyczyński (2001) W. A. Woyczyński, “Lévy processes in the physical sciences,” in Lévy Processes: Theory and Applications, edited by O. E. Barndorff-Nielsen, S. I. Resnick, and T. Mikosch (Birkhäuser Boston, Boston, MA, 2001) pp. 241–266.
- Pereira et al. (2004) E. Pereira, J. M. G. Martinho, and M. N. Berberan-Santos, Phys. Rev. Lett. 93, 120201 (2004).
- Novikov et al. (2005) D. S. Novikov, M. Drndic, L. S. Levitov, M. A. Kastner, M. V. Jarosz, and M. G. Bawendi, Phys. Rev. B 72, 075309 (2005).
- Barthelemy et al. (2008) P. Barthelemy, J. Bertolotti, and D. S. Wiersma, Nature 453, 495 (2008).
- Augello et al. (2010) G. Augello, D. Valenti, and B. Spagnolo, Eur. Phys. J. B 78, 225 (2010).
- Luryi et al. (2012) S. Luryi, O. Semyonov, A. Subashiev, and Z. Chen, Phys. Rev. B 86, 201201 (2012).
- Semyonov et al. (2012) O. Semyonov, A. V. Subashiev, Z. Chen, and S. Luryi, J. Lumin. 132, 1935 (2012).
- Briskot et al. (2014) U. Briskot, I. A. Dmitriev, and A. D. Mirlin, Phys. Rev. B 89, 075414 (2014).
- Subashiev et al. (2014) A. V. Subashiev, O. Semyonov, Z. Chen, and S. Luryi, Phys. Lett. A 378, 266 (2014).
- Vermeersch et al. (2014) B. Vermeersch, A. M. S. Mohammed, G. Pernot, Y. R. Koh, and A. Shakouri, Phys. Rev. B 90, 014306 (2014).
- Mohammed et al. (2015) A. M. S. Mohammed, Y. R. Koh, B. Vermeersch, H. Lu, P. G. Burke, A. C. Gossard, and A. Shakouri, Nano Letters 15, 4269 (2015).
- Vermeersch et al. (2015a) B. Vermeersch, J. Carrete, N. Mingo, and A. Shakouri, Phys. Rev. B 91, 085202 (2015a).
- Vermeersch et al. (2015b) B. Vermeersch, A. M. S. Mohammed, G. Pernot, Y. R. Koh, and A. Shakouri, Phys. Rev. B 91, 085203 (2015b).
- Gattenlöhner et al. (2016) S. Gattenlöhner, I. V. Gornyi, P. M. Ostrovsky, B. Trauzettel, A. D. Mirlin, and M. Titov, Phys. Rev. Lett. 117, 046603 (2016).
- Gutman et al. (2016) D. B. Gutman, I. V. Protopopov, A. L. Burin, I. V. Gornyi, R. A. Santos, and A. D. Mirlin, Phys. Rev. B 93, 245427 (2016).
- Dubkov et al. (2008) A. A. Dubkov, B. Spagnolo, and V. V. Uchaikin, International Journal of Bifurcation and Chaos 18, 2649 (2008).
- Zaburdaev et al. (2015) V. Zaburdaev, S. Denisov, and J. Klafter, Rev. Mod. Phys. 87, 483 (2015).
- Nolan (2017) J. P. Nolan, “Bibliography on stable distributions, processes and related topics,” http://academic2.american.edu/ jpnolan/stable/stable.html (2017), [Online].
- Grabert and Weiss (1984) H. Grabert and U. Weiss, Phys. Rev. Lett. 53, 1787 (1984).
- Weron (1996) R. Weron, Stat. Probab. Lett. 28, 165 (1996).
- Chambers et al. (1976) J. M. Chambers, C. L. Mallows, and B. W. Stuck, J. Amer. Statist. Assoc. 71, 340 (1976).
- Dubkov and Spagnolo (2007) A. Dubkov and B. Spagnolo, Acta Phys. Pol. B 38, 1745 (2007).
- Valenti et al. (2014) D. Valenti, C. Guarcello, and B. Spagnolo, Phys. Rev. B 89, 214510 (2014).
- Guarcello et al. (2016) C. Guarcello, D. Valenti, A. Carollo, and B. Spagnolo, J. Stat. Mech.: Theory Exp. 2016, 054012 (2016).
- Dybiec et al. (2006) B. Dybiec, E. Gudowska-Nowak, and P. Hänggi, Phys. Rev. E 73, 046104 (2006).
- Dybiec et al. (2007) B. Dybiec, E. Gudowska-Nowak, and P. Hänggi, Phys. Rev. E 75, 021109 (2007).
- Dubkov et al. (2009) A. A. Dubkov, A. L. Cognata, and B. Spagnolo, J. Stat. Mech.: Theory Exp. 2009, P01002 (2009).
- Chechkin et al. (2005) A. V. Chechkin, V. Y. Gonchar, J. Klafter, and R. Metzler, Europhys. Lett. 72, 348 (2005).
- Chechkin et al. (2007) A. V. Chechkin, O. Y. Sliusarenko, R. Metzler, and J. Klafter, Phys. Rev. E 75, 041101 (2007).
- Kramers (1940) H. Kramers, Physica 7, 284 (1940).
- Fulton and Dunkleberger (1974) T. A. Fulton and L. N. Dunkleberger, Phys. Rev. B 9, 4760 (1974).
- Guarcello et al. (2017) C. Guarcello, D. Valenti, B. Spagnolo, V. Pierro, and G. Filatrella, Nanotechnology 28, 134001 (2017).
- Guarcello et al. (2015) C. Guarcello, D. Valenti, and B. Spagnolo, Phys. Rev. B 92, 174519 (2015).
- Giubileo et al. (2017) F. Giubileo, N. Martucciello, and A. D. Bartolomeo, Nanotechnology 28, 410201 (2017).
- Spagnolo et al. (2017) B. Spagnolo, C. Guarcello, L. Magazzú, A. Carollo, D. Persano Adorno, and D. Valenti, Entropy 19 (2017), 10.3390/e19010020.
- Lee and Lee (2017) G.-H. Lee and H.-J. Lee, arXiv preprint arXiv:1709.09335 (2017).
- Prudnikov et al. (1998) A. P. Prudnikov, Y. Brychkov, and O. I. Marichov, Integrals and Series, Vol. 2 (Gordon and Breach, India, 1998).